Very high current relativistic electron accelerators are presently required for free electron lasers as discussed in Sprangle, Smith, and Granatstein, Free Electron Lasers and Stimulated Scattering from Relativistic Electron Beams, N.R.L. Memorandum Report 3911 (1978) and Caponi, Munch, and Boehmer, Optimized Operation of a Free Electron Laser, etc., J. Quantum Elec. (to be published). Intense relativistic charged particle beams are also useful as intense gamma ray sources and for nuclear physics research.
Relativistic electron accelerators normally carry electron beam currents measured in micro- or milliamperes. A major obstacle to the achievement of higher currents is beam blowup due to charge instability at the time of injection of electrons.
The onset of beam blowup may be estimated by reference to a characteristic time which is the inverse of the beam plasma frequency. A characteristic blowup time, t.sub.b, may be approximated by EQU (ct.sub.b).sup.2 =2.pi..gamma..sup.3 /nr.sub.e ( 1)
where c is the velocity of light, n the number density of electrons, and r.sub.e is the classical electron radius which is about 2.6.times.10.sup.-13 cm. The relativistic time dilation factor, .gamma., is defined by EQU .gamma.=(1-V.sup.2 /c.sup.2).sup.-1/2 ( 2)
where V is the beam velocity at injection.
The injection current I, with cross-sectional dimension, a, is EQU I=.pi.eVna.sup.2 ( 3)
e is the charge of the electron.
Beam blowup occurs when the characteristic blowup time, t.sub.b, exceeds the residence time of the electrons in some characteristic acceleration length, l; that is, t.sub.b exceeds l/2 V. Consequently the injection current is limited by EQU I&lt;&lt;8.pi..sup.2 (a/l).sup.2 .gamma..sup.3 (ec/r.sub.e) (V/c).sup.3 ( 4)
Taking l to be of the order of magnitude of 1 m. and a about 1 cm. gives EQU I&lt;&lt;10.sup.4 .gamma..sup.3 (V/c).sup.3 amps. (5)
If the electrons are injected with energies of a few kilovolts, corresponding to V/c about 0.1, then space charge instability limits the injection current to be much smaller than a few amperes. The estimate in Eq. (5) also shows that when the beam becomes relativistic with (V/c) close to unity and .gamma. large, the space charge limit on the beam may be tens of kiloamperes.
For a more careful estimate of the limiting effect of space chage instability it is desirable to consider a specific geometry such as a conventional betatron.
In order to describe the particle orbits of a toroidal electron beam in a conventional betatron consider the local coordinates (x,y) or (r,.theta.) as illustrated in FIG. 1. The toroidal direction is indicated by z. If the velocity of the beam in the toroidal direction is large, V&gt;&gt;v.sub.x,v.sub.y, the equations of motion for an electron with mass m are approximately ##EQU1## where R is the major radius of the beam axis. EQU E.sub.r =-2Ner/a (8)
and EQU B.theta.=-2NeVr/ca (9)
are the self-fields of the beam assumed to have uniform density. Also EQU B.sub..theta. =.beta.E.sub.r ( 10)
where EQU .beta.=V/c. (11)
B.sub.y is a vertical magnetic field which is the important magnetic field for a conventional betatron. B.sub.x is also important in a conventional betatron.
The self-field terms are ##EQU2## where .omega..sub.p is the beam plasma frequency given by EQU .omega..sub.p.sup.2 =4.pi.nc.sup.2 r.sub.e /.gamma. (13)
Assuming the betatron magnetic fields to be approximated near the z-axis by ##EQU3## where s is the same fixed power, then the vanishing of the curl of the B-field implies EQU B.sub.x =sB(y/R) (15)
The equations of motion then simplify to EQU (d.sup.2 x/dt.sup.2)-.OMEGA..sup.2 x+.OMEGA..sub.y.sup.2 (1-s)x=0 (16) EQU (d.sup.2 y/dt.sup.2)-.OMEGA..sup.2 y+.OMEGA..sub.y.sup.2 sy=0 (17)
where EQU .OMEGA..sup.2 =.OMEGA..sub.p.sup.2 /2.gamma..sup.2 ( 18a)
and EQU .OMEGA.y=eB/.gamma.mc=V/R (18b)
Except for the .OMEGA..sup.2 terms these are the standard betatron equations. The Betatron fields B.sub.x,B.sub.y produce focusing and the self-fields produce the de-focusing terms -.OMEGA..sup.2 x and -.OMEGA..sup.2 y. The condition for orbit stability is EQU .OMEGA..sub.y.sup.2 /2&gt;.OMEGA..sup.2 ( 19)
or EQU n&lt;.gamma.B.sup.2 /(4.pi.mc.sup.2) (20)
Then the current satisfies EQU I&lt;.gamma..sup.3 (V/c).sup.3 (ec/r.sub.o)(a/R).sup.2 ( 21)
which, for R about 1 m. and beam area about one cm.sup.2 becomes approximately .gamma..sup.3 (V/c).sup.3 amperes. In the University of Illinois betatron the electrons were injected at about 100 Kev. The space charge limit on the current would therefore be of the order of a hundred milliamperes, according to Eq. (21), consistent with experience.
Inductive charging is a known technique for confinement of non-neutral plasmas with high charge densities. The technique, in principle, involves the introduction of charged particles into a continuous region of cylindrical cross-section. A uniform, axial magnetic field, is then created in the containment region by causing current to increase in a solenoid surrounding the cylindrical region. As the magnetic field increases, the charged particles are forced to move toward the center of the cylindrical confinement region, thereby creating an inner cylindrical of high charge density.
Inductive charging of a torus has been discussed by Clark, Korn, Mondelli, and Rostoker, Phys. Rev. Lett. 37, 592 (1976).
In accordance with the present invention the injection problem for accelerators is solved by the method of inductive charging. This method involves a longitudial magnetic field B.sub.z and thermionic electron injectors near the edge of the accelerator. Electrons are injected when the magnetic field B.sub.z is low. The field is then increased thereby compressing the electrons into a small cylindrical region at the center of the accelerator.
The charge limit may be evaluated from the equations of motion with B.sub.x, B.sub.y, eliminated but with the longitudinal field B.sub.z, included. The initial concern is with trapping rather than accelerating electrons; V is therefore set to zero. The equations of motion are EQU (d.sup.2 x/dt)-.OMEGA..sup.2 x+.OMEGA..sub.z (dy/dt)=0 (22) EQU (d.sup.2 y/dt.sup.2)-.OMEGA..sup.2 y-.OMEGA..sub.z (dx/dt)=0 (23)
where EQU .OMEGA..sub.z =eB.sub.z /mc. (24)
The condition for stability is .OMEGA..sub.z.sup.2 &gt;4.OMEGA..sup.2 or EQU n.ltoreq.B.sub.z.sup.2 /(8.pi.mc.sup.2) (25)
For B.sub.z about one kilogauss the space charge limit is n=6.times.10.sup.11 cm.sup.-3, corresponding to a current in a beam of about 1 cm radius after acceleration of about 10 kiloamperes.
The method of injection by inductive charging therefore promises an improvement of 4-6 orders of magnitude in beam current in electron accelerators.
Use of inductive charging in an accelerator having a toroidal geometry in fact requires a large beam current during injection. A lower limit on the beam current arises from the inherent small inhomogeneity of the magnetic field within a toroidal solenoid. The small inhomegeneity may be calculated, for example, by following Bleaney, B. I. and Bleaney B., Electricity and Magnetism, London, Oxford University Press, 1957, pp. 133-134. The magnetic field near the z-axis is EQU B.sub.z =B.sub.zo R(R-x).sup.-1 .apprxeq.B.sub.zo (1+x/R) (26)
where use is made of the fact that x/R is small close to the z-axis and a rectangular coil is assumed. In Eq. (26) B.sub.zo is the field at the origin of the x-axis and the positive x-direction is taken to be toward the major axis of the torus.
The effect of the (1+x/R) factor in Eq. (26) is to modify Eqs. (22) and (23) by replacing .OMEGA..sub.z in those equations by .OMEGA..sub.z (1+x/R) where .OMEGA..sub.z is now defined by Eq. (24) with B.sub.z replaced by B.sub.zo.
Approximate solutions to the modified Eqs. (22) and (23) are obtainable by a formal expansion in 1/R. Also, it is assumed that .OMEGA./.OMEGA..sub.z is small because .OMEGA. is proportional to the current density and the existence of a lower limit on the current density is being shown.
The lowest order (R=.infin.) solutions to Eqs. (22) and (23) are EQU x=(X-.rho. cos .alpha.) cos .omega.t-(Y-.rho. sin .alpha.) sin .omega.t+.rho. cos (.OMEGA..sub.z t+.alpha.) (27) EQU y=(Y-.rho. sin .alpha.) cos .omega.t+(X-.rho. cos .alpha.) sin .omega.t+.rho. sin (.OMEGA..sub.z t+.alpha.) (28)
where X and Y are the initial values of x and y and EQU .omega.=.OMEGA..sup.2 /.OMEGA..sub.z &lt;&lt;.OMEGA..sub.z ( 29)
Eqs. (27) and (28) describe a rapid circular gyration of radius .rho. and frequency .OMEGA..sub.z about a guiding center. The guiding center drifts around the z-axis in a circle of radius EQU R.sub.d =[(X-.rho. cos .alpha.).sup.2 +(Y-.rho. sin .alpha.).sup.2 ].sup.1/2( 30)
which may be much larger if either X or Y is substantially larger than .rho.. The velocity of the rapid circular gyration is given by EQU V.sub.perp =.rho..OMEGA..sub.z ( 31)
The 1/R corrections to Eq. (27) and (28), denoted by x.sub.1, y.sub.1, are obtained by substituting the lowest order solutions in the non-linear terms of Eqs. (22) and (23) modified according to Eq. (26) and solving the resulting linear equations for x.sub.1, y.sub.1. The non-linear terms are now "driving terms" for the new equations and will consist in general of constants and oscillating terms having as frequencies sums, differences, and multiples of the frequencies .OMEGA. and .OMEGA..sub.z. Ignoring the oscillating terms, which result in small perturbations to the orbit given by Eqs. (27) and (28) and keeping only the constant term results in the equations EQU (d.sup.2 /dt.sup.2)x.sub.1 -.OMEGA..sup.2 x.sub.1 +.OMEGA..sub.z (d/dt)y.sub.1 =-1/2.rho..sup.2 .OMEGA..sub.z.sup.2 /R (32) EQU (d.sup.2 /dt.sup.2)y.sub.1 -.OMEGA..sup.2 y.sub.1 -.OMEGA..sub.z (d/dt)x.sub.1 =0 (33)
with the initial conditions EQU x.sub.1 =y.sub.1 =(d/dt)x.sub.1 =(d/dt)y.sub.1 =0 (34)
Inspection of Eq. (32) shows that the constant "driving term" on the right is eliminated by the substitution EQU x.sub.1 .fwdarw.x.sub.1 +.rho..sup.2 .OMEGA..sub.z.sup.2 /(2R.OMEGA..sup.2).tbd.x.sub.1 +V.sup.2.sub.perp /2R.OMEGA..sup.2 ( 35)
corresponding to a shift in the guiding center in the x-direction by an amount EQU x.sub.o =V.sup.2.sub.perp /2R.OMEGA..sup.2 ( 36)
In order to have a stable beam it is necessary that the guiding center shift be less than about the beam radius a. Use of the defining Eqs. (13) and (18a), with .gamma. taken as unity to correspond to injection conditions then gives as a stability condition on N, the line density of electrons EQU N=.pi.a.sup.2 n&gt;1/4(a/R)mV.sup.2.sub.perp /e.sup.2 ( 37)
Assuming that a is about 1 cm., R about 1 m., and (1/2)mV.sup.2.sub.perp, the injection energy, is about 15 Kev before B.sub.z increases, corresponding to about 100 Kev after B.sub.z is increased, gives as a result that N must be greater than about 10.sup.9 cm.sup.-1. After acceleration the minimum current is EQU I.sub.e =Nec&gt;10A. (38)
Higher injection energies will, of course, require larger minimum currents.
The limit given by Eq. (38) emphasizes the difference between an inductively charged accelerator and a conventional cyclic accelerator. In known cyclic accelerators there is never a magnetic field B.sub.z. This is because a toroidal magnetic field B.sub.z is always slightly inhomogeneous as in Eq. (26). The inhomegeneity would always cause particles to drift into the accelerator wall in a short time if the current or particle density were very small as it is in a conventional cyclic accelerator. However a large particle density introduces a new physical effect, namely, the circular particle drift due to the self-electric field of the beam. This drift can cancel the drift caused by the magnetic field inhomogeneity if the electron density or final beam current is sufficiently large.
Toroidal magnetic fields are known in connection with tokamaks where they also give rise to a drift which must be corrected. However, the method of correction described in connection with the present invention would not be applicable to a tokamak because tokamaks contain neutral plasma and there can be no self electric field for the plasma.
Eqs. (22) and (23) describe the particle orbits before acceleration in the z direction. When acceleration takes place .OMEGA..sup.2 decreases as .gamma. increases, as shown in Eq. 18(a), and eventually becomes too small to correct the toroidal drift. However when .gamma. is large, space charge is no longer a problem and there is no need to have the toroidal field B.sub.z which is introduced only to control space charge at low energy when .gamma. is close to unity. At large .gamma. the toroidal field may be permitted to decay so that drift from the toroidal field inhomogeneity need not be compensated.